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6 THE FREDHOLM INTEGRAL EQUATION 345
by (8.5.3). But the left-hand side is (U<p \ <p), and this violates the assumption
that U is a positive operator.
Remark now that for any finite number of eigenvalues A* (1 < k < ri)
n _
KW(J, 0 = KO, t) — j) 4 <Pk(f)9k(t) is the kernel function of a positive oper-
fc=i
ator £/„ , for we have |
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but the right-hand side of that equation can be written (Ug\g) with
n
g =/- £ (/| <pfc)<p*, as is readily verified, hence is positive by assumption.
jt=i
Therefore, by (5.3.1) it follows from Kn(s, s)^0 that the series £ An |<pB(j)|2
n
is convergent, and we have ]T !„ I^C?)!2 < K(j, j) for all sel. By Cauchy-
n
Schwarz, we conclude that
0
(11.6.7.1) E
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for all Os, 0 e I x I. Hence, as K(^ t) is bounded in I, f or fixed s e I, the series
) *s uniformly convergent for /el. By (11.6.6), (8.7.8), and |
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(8.5.3), we conclude that £ ln <pn(s)(pn(t) = K(s, t) for all (s, t) e I x I since
n
2 |
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t ->• K(5", 0 ~~ X ^" (Pn(s)(Pn(f) is continuous in I and its integral in I is 0. In
n
particular, we have K(s, s) = Ys^n\(Pn(s)\2l by Dini's theorem (7.2.2) the
n
series £^n|<pnCs)|2 l$ therefore uniformly convergent in I, and (11.6.7.1)
n ._____
proves that the series J] An (pn(s)(pn(t) is absolutely and uniformly convergent
n
in I x 1, which ends the proof.
jRtfWtfrfcs
(11.6.8) The result (11.6.7) is still true when we only suppose that t/has a
finite number of eigenvalues vk < 0 (1 < k < m). For (11.5.7(c)) shows then
that in the space F^+1, orthogonal supplement of E(v1) +-----h E(vm) in G,
the restriction of the operator U is positive, and we apply (11.6.7) to that
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